GL: 


UC-NRLF 


/On  the  deter-N 
mination  of 
elliptic  or- 
bite  from  3 
complete  ob- 
servations. 

N.A.S.  4.  8th 
Mem. 


Students' 

.  Observatori 


NATIONAL    ACADEMY    OF    SCIKNCES. 


V  01,.     TV. 


EIGHTH    MEMOIR. 


ON  THE  DETERMINATION -OF  ELLIPTIC  ORBITS  FROM  THREE 
COMPLETE  OBSERVATIONS. 


NATIONAL    ACADEMY    OF    SCIENCES. 


VOL.     IV. 


EIGHTH   MEMOIR. 


ON  THE  DETERMINATION  OF  ELLIPTIC  ORBITS  FROM  THREE 
COMPLETE  OBSERVATIONS. 


79 


ON  THE  DETERMINATION  OF  ELLIPTIC  ORBITS  FROM  THREE  COMPLETE 

OBSERVATIONS. 


By  J.  WILLARD  GIBBS. 


The  determination  of  an  orbit  from  three  complete  observations  by  the  solution  of  the  equa- 
tions which  represent  elliptic  motion  presents  so  great  difficulties  in  the  general  case,  that  in  the 
first  solution  of  the  problem  we  must  generally  limit  ourselves  to  the  case  in  which  the  intervals 
between  the  observations  are  not  very  long.  In  this  case  we  substitute  some  comparatively  simple 
relations  between  the  unknown  quantities  of  the  problem,  which  have  an  approximate  validity  for 
short  intervals,  for  the  less  manageable  relations  which  rigorously  subsist  between  these  quantities. 
A  comparison  of  the  approximate  solution  thus  obtained  with  the  exact  laws  of  elliptic  motion 
will  always  afford  the  means  of  a  closer  approximation,  and  by  a  repetition  of  this  process  we  may 
arrive  at  any  required  degree  of  accuracy. 

It  is  therefore  a  problem  not  without  interest  —  it  is,  in  fact,  the  natural  point  of  departure  in 
the  study  of  the  determination  of  orbits  —  to  express  in  a  manner  combining  as  far  as  possible  sim- 
plicity and  accuracy  the  relations  between  three  positions  in  an  orbit  separated  by  small  or  mod- 
erate intervals.  The  problem  is  not  entirely  determinate,  for  we  may  lay  the  greater  stress  upon 
simplicity  or  upon  accuracy  ;  we  may  seek  the  most  simple  relations  which  are  sufficiently  accurate 
to  give  us  any  approximation  to  an  orbit,  or  we  may  seek  the  most  exact  expression  of  the  real 
relations,  which  shall  not  be  too  complex  to  be  serviceable. 

DERIVATION   OF    THE   FUNDAMENTAL   EQUATION. 

The  following  very  simple  considerations  afford  a  vector  equation,  not  very  complex  and  quite 
amenable  to  analytical  transformation,  which  expresses  the  relations  between  three  positions  in 
an  orbit  separated  by  small  or  moderate  intervals,  with  an  accuracy  far  exceeding  that  of  the 
approximate  relations  generally  used  in  the  determination  of  orbits. 

If  we  adopt  such  a  unit  of  time  that  the  acceleration  due  to  the  sun's  action  is  nnity  at  a 
unit's  distance,  and  denote  Lhe  vectors*  drawn  from  the  SUB  to  the  body  in  its  three  positions  by 

*  Vectors,  or  directed  quantities,  will  be  represented  iu  this  paper  by  German  capitals.  The  following  notations  will 
be  used  in  connection  with  them. 

The  sign  =  denotes  identity  iu  direction  as  well  as  length. 

The  sign  -f-  denotes  geometrical  addition,  or  what  is  called  composition  iu  mechanics. 

The  sign  —  denotes  reversal  of  direction,  or  composition  after  reversal. 

The  notation  $l-$$  denotes  the  product  of  the  lengths  of  the  vectors  and  the  cosine  of  the  angle  which  they 
include.  It  will  be  called  the  direct  product  of  $[  and  §3.  If  x,  y,  z  are  the  rectangular  components  of  $(,  and  x1,  y', 
i'  those  of  S3, 


Sl-Vl  may  be  written  W  and  called  the  square  of  fl. 

The  notation  SlXg  will  be  used  to  denote  a  vector  of  which  the  length  is  the  product  of  the  lengths  of  Jl  and  93 
and  the  sine  of  the  angle  which  they  include.     Its  direction  is  perpendicular  to  Jl  and  2J,  and  on  that  side  on  which 
H.  Mis.  597  -  6  81 


M789809 


82  MEMOlltS  OF  THE  NATIONAL  ACADEMY  OF  SCIENCES. 

tt],  Wf,  9t],  and  the  lengths  of  these  vectors  (the  hHiix-i'ntric  distances)  by  ri,  r,,  r3,  the  acceler- 

OJ  kU  01 

ations  corresponding  to  the  three  positions  will  be  represented  by—    j  .  _  Z£  ,  _  -£.      Now  the 

motion  between  the  positions  considered  may  be  expressed  with  a  high  degree  of  accuracy  by  an 
equation  of  the  form 


having  five  vector  constants.  The  actual  motion  rigorously  satisfies  six  conditions,  viz.,  if  we  write 
TJ  for  the  interval  of  time  between  the  first  and  second  positions,  and  TI  for  that  between  the  second 
and  third,  and  set  t=0  for  the  second  position, 

for  t=—  r,, 

a,_su  d»8t         JR,. 

dt*  ~  -j?' 
for  f=0, 

aj     01  w  «H         y\2  . 

a?=  ~T?» 

for  tssTi, 

m  _  g\  «  nl  lK3 

«//J  "  "??• 

We  may  therefore  write  with  a  high  degree  of  approximation  : 

Ki=8 
«,=« 


•  rotation  from  ft  to  SJ  appears  connter-clock-wise.    It  will  be  called  the  ikew  product  of  Jl  and  SJ.     If  the  rectan- 
gular components  of  ft  and  <B  are  x,  y,  t,  and  x1,  y1,  t1,  those  of  fix 8)  will  be 

0 

yz'—zy',  zx'—xz',  ry—yr'. 

The  notation  (JISC)  denotes  the  volume  of  the  parallelepiped  of  which  three  edges  are  obtained  by  laying  off 
the  vectors  JI,  8,  and  G  from  any  same  point,  which  volume  is  to  be  taken  positively  or  negatively,  according  as 
the  vector  G  falls  on  the  side  of  the  plane  containing  ?l  and  ^,  on  which  :i  rotation  from  *l  to  5)  appears  counter- 
clock-wise,  or  on  the  other  side.  If  the  rectangular  components  of  ft,  *l,  and  G  are  x,  y,  z ;  x1,  y7,  z1 ;  and  x",  y",  t", 


(*»<&)  = 


x  y  z 
x1  y'  .-' 
x"  y"  t" 


It  follows,  from  the  above  definitions,  that  for  any  vectors  ft,  8,  and  G 

«•»=»•«,  *x»=-»x»,  («$G)=(^G«)=(G*^)= 

and 

(»8G)=«-(»  x  G)  =  »-(G  X  «)=<?•(*  X  *); 


also  that  ft-8,  <lx8,  »re  distrihutive  functions  of  ft  and  3),  »"'!  (fl'HG)  a  dintributive  function  of  ft,  8,  and  G, 
for  example,  that  if  ft= 


and  to  for  $  and  G. 

The  notation  (ft»\G)  '•  identical  with  that  of  Lagrange  in  tho  M/caniqtie  A*alyti<ptt,  exci-|it  that  tin-re  its  use  is 
limited  to  unit  vectors.  The  signification  of  ft  X  4^  '»  dourly  r.-lat.-d  to,  hut  n,.t  id.-ntu-al  with,  that  .if  Ih.-  nota- 
tion [r,r.]  commonly  used  to  denote  the  double  area  of  a  tuan^l.  il-  t.  i  n.iu.  .1  l>v  two  |HiHitiuini  in  un  orbit. 


MEMOIRS  OF  THE  NATIONAL  ACADEMY  OF  SCIENCES.  83 

From  these  six  equations  the  five  constants  SI,  S,  (i,  25,  £  may  be  eliminated,  leaving  a  single 
equation  of  the  form 

where 

f.  •»•_ 

A,=- 


This  we  shall  call  our  fundamental  equation.    In  order  to  discuss  its  geometrical  signification, 
let  us  set 


so  that  the  equation  will  read 

0.  (3) 


This  expresses  that  the  vector  »2S2  is  the  diagonal  of  a  parallelogram  of  which  niSRi  and  n39R3  are 
sides.    If  we  multiply  by  S3  and  by  9l|  ,  in  sfcetc  multiplication,  we  get 


=0,  (4) 

whence 

ii\     .  .  v»i  ,n     .  .  ,n  ill     .  .  iii 

(5) 


Our  equation  may  therefore  be  regarded  as  signifying  that  the  three  vectors  Sfti,  SR2,  $3  lie  in  one 
plane,  and  that  the  three  triangles  determined  each  by  a  pair  of  these  vectors,  and  usually  de- 
noted by  [*V3],  [rir3],  [n^],  are  proportional  to 


Since  this  vector  equation  is  equivalent  to  three  ordinary  equations,  it  is  evidently  sufficient 
to  determine  the  three  positions  of  the  body  in  connection  with  the  conditions  that  these  positions 
must  lie  upon  the  lines  of  sight  of  three  observations.  To  give  analytical  expression  to  these 
conditions,  we  may  write  d,  &2,  (S3  for  the  vectors  drawn  from  the  sun  to  the  three  positions  of 
the  earth  (or,  more  exactly,  of  the  observatories  where  the  observations  have  been  made),  gi  ,  g2,  g, 
for  unit  vectors  drawn  in  the  directions  of  the  body,  as  observed,  and  pi,  pi,  p3  for  the  three 
distances  of  the  body  from  the  places  of  observation.  We  have  then 


(6) 

By  substitution  of  these  values  our  fundamental  equation  becomes 


where  PI,  PI,  P3,ri,rt,r3  (the  geocentric  and  heliocentric  distances)  are  the  only  unknown  quanti- 
ties.   From  equations  (6)  we  also  get,  by  squaring  both  members  in  each, 


(8) 

by  which  the  values  of  rt,  r2,  r3  may  be  derived  from  those  of  pi,  pz,  p-j,  or  vice  versa.    Equations 
(7)  ami  (8),  which  are  equivalent  to  six  ordinary  equations,  are  sufficient  to  determine  the  six 


84  MIIMOIKS  OF  TI1K   NATIONAL  AUADK.MY  OK  SCIENCES. 

quantities  rlt  rf,  r5,  pi,  /a,,  /3j;  or,  if  we  suppose  the  \  dues  of  n,  ra,  r3  in  terms  of  pi,  p,,  p^  to  be 
substituted  in  equation  (7),  we  have  a  single  vector  equation,  from  which  we  may  determine  the 
three  geocentric  distances  pi,  pi,  p3. 

It  remains  to  be  shown,  fust,  how  the  numerical  solution  of  the  equation  may  be  performed, 
and,  secondly,  how  such  an  approximate  solution  of  the  actual  problem  may  furnish  the  basis  of 
a  closer  approximation. 

SOLUTION  OP  THB  FUNDAMENTAL  EQUATION. 

The  relations  with  which  we  have  to  do  will  be  rendered  a  little  more  simple  if  instead  of  each 
geocentric  distance  we  introduce  the  distance  of  the  body  from  the  foot  of  the  perpendicular  from 
the  Min  upon  the  line  of  sight.  If  we  set 


equations  (8)  become 

ri^qf+tf,  rf=qf+pf,  r3*=q3*+p3*.  (11) 

Let  us  also  set,  for  brevity, 

®»=-(i-^)(e2+/os&),       «*-*(i+§)(fc+/>*).     (12) 

Then  S,,  2,,  g,  may  be  regarded  as  functions  respectively  of  PI,  p»,  pa,  therefore  of  ji,  «3,  q3, 
and  if  we  set 

g'=g-',  «"=?%  &"=f\  (13) 

aqi  dq,  dq3  ' 

and 

s=e,+£2-i-S3,  (U) 

we  shall  have 

<i2=£'dql  +  £"eiq1+£"'dq3.  (15) 

To  determine  the  value  of  g',  we  get  by  differentiation 


But  by  (11) 

*"!_  ?1  .    _. 

^-r," 
Therefore 


(IS) 


Now  if  any  value*  of  ?,,  q,,  qt  (either  assumed  orolitain.-d  hy  a  previous  a|)pn>xiinatioii)  «ivi- 
•  certain  reM.lual  :  (.vl.irh  woul.l  I,,,  /.,-n,  if  tin-  raldefl  of  ,,,,  ,,.,  ,h  sati.slie.l  the  fuu.lameutal 
equation),  and  we  wish  to  tin.l  the  eornvtions  J,h ,  J,,:.  Jft(  w|,i,.|,  must  b,-  a.l.l.-d  to  ,,, .  ,/..  ,, 


"^rjO^+r,  ) 

t   "3 


MEMOIRS  OF  THE  NATIONAL  ACADEMY  OF  SCIENCES.  85 

to  reduce  the  residual  to  zero,  we  may  apply  equation  (15)  to  these  finite  differences,  and  will  have 
approximately,  when  these  differences  are  not  very  large, 


^.  (19) 

This  gives* 

(23''®"')  (e<s'"£')  (®g'e") 

J«'      (S'S'^}        ^=-(@/©"g'")        Jfc=-(Wl^y 

From  the  corrected  values  of  q\,qz,  q3  we  may  calculate  a  new  residual  3,  and  from  that  determine 
another  correction  for  each  of  the  quantities  q\,  q%,  q3. 

It  will  sometimes  be  worth  while  to  use  formulas  a  little  less  simple  for  the  sake  of  a  more 
rapid  approximation.     Instead  of  equation  (19)  we  may  write,  with  a  higher  degree  of  accuracy, 


,  (21) 

where 


(22) 


S'"  = 


It  is  evident  that  1"  is  generally  many  times  greater  than  S'  or  2'",  the  factor  .B2,  in  the  case  of 
equal  intervals,  being  exactly  ten  times  as  great  as  A^B,  or  A3B3.  This  shows,  in  the  first  place, 
that  the  accurate  determination  of  Aq^  is  of  the  most  importance  for  the  subsequent  approxi- 
mations. It  also  shows  that  we  may  attain  nearly  the  same  accuracy  in  writing 


222  (23) 

We  may,  however,  often  do  a  little  better  than  this  without  using  a  more  complicated  equation. 
For  J'+I'"  may  be  estimated  very  roughly  as  equal  to  \\".  Whenever,  therefore,  Aqv  and  Aq, 
are  about  as  large  as  Jg2  ,  as  is  often  the  case,  it  may  be  a  little  better  to  use  the  coefficient  &- 
instead  of  J  in  the  last  term. 

For  Jg2  ,  then,  we  have  the  equation 


.        (24) 
is  easily  computed  from  the  formula 

which  may  be  derived  from  equations  (18)  and  (22). 

The  quadratic  equation  (24)  gives  two  values  of  .the  correction  to  be  applied  to  the  position  of 
the  body.  When  they  are  not  too  large,  they  will  belong  to  two  different  solutions  of  the  problem, 
generally  to  the  two  least  removed  from  the  values  assumed.  But  a  very  large  value  of  Aq%  muse 
not  be  regarded  as  affording  any  trustworthy  indication  of  a  solution  of  the  problem.  In  the 
majority  of  cases,  we  only  care  for  one  of  the  roots  of  the  equation,  which  is  distinguished  by 
being  very  small,  and  which  will  be  most  easily  calculated  by  a  small  correction  to  the  value  which 
we  get  by  neglecting  the  quadratic  terni.t 

*  These  equations  are  obtained  by  taking  the  direct  products  of  both  members  of  the  preceding  equation  with 
£"  X  £'",  £'"  X  2',  and  2'  x  2",  respectively.  See  foot-note  on  page  81. 

tin  the  case  of  Swift's  comet  (V,  1880),  the  writer  found  by  the  quadratic  equation  —.247  and  —.116  for  cor- 
rections of  the  assumed  geocentric  distance  .250.  The  first  of  these  numbers  gives  au  approximation  to  the  position  ' 
of  the  earth;  the  second  to  that  of  the  comet,  viz.,  the  geocentric  distance  .134  instead  of  the  true  value  .1333.  The 
coefficient  -fa  was  used  in  the  quadratic  equation;  with  the  coefficient  \  the  approximations  would  not  be  quite  so 
good.  The  value  of  the  correction  obtained  by  neglecting  the  quadratic  term  was  .070,  which  indicates  that  the 
approximations  (in  this  very  critical  ease)  would  l>e  quite  tedious  without  the  use  of  the  quadratic  term. 


H«;  MKMOIKS  oi-  THE  NATIONAL  ACADEMY  OF  SCIENCES. 

When  a  comet  is  somewhat  near  the  earth  we  may  make  use  of  the  fact  that  the  earth's  orbit 
is  one  solution  of  the  problem,  i.  e.,  that  —f>t  is  one  value  of  Jg2,  to  save  the  trifling  labor  of  com- 
puting the  value  of  (2"2'"2')-  For  it  i*  i-vident  from  the  theory  of  equations  that  if  —  p1  and  z 
are  the  two  roots, 

_(2'2  [€  (22'"2') 

Pt~z~l(l"2"/2l)  ~f(3"2'""2'j°  • 

Eliminating  (1"2'"£'),  we  have 

(p,- 
whence 


Now  —  -£        m]  is  the  val°e  °f  ^9»>  wnicl1  we  obtain  if  we  neglect  the  quadratic  term  in  equa- 
tion (24).    If  we  call  this  value  [dq,],  we  have  for  the  more  exact  value* 

Jft-._M«!jL 

9  H.L£*]  <26> 

p> 

The  quantities  dq\  and  Jg3  might  be  calculated  by  the  equations 

2 


But  a  little  examination  will  show  that  the  coefficients  of  Aqf  in  these  equations  will  not  generally 
have  very  different  values  from  the  coefficient  of  the  same  quantity  in  equation  (24).  We  may 
therefore  write  with  sufficient  accuracy 

J$i=[Jgi]+J«,-M««J»  Aq3=[Aq3]  +  Jqt-[Aq2],  (28) 

where  [dq\],  [^<7z],  l-^ft]  denote  the  values  obtained  from  equations  (20). 

In  making  successive  corrections  of  the  distances  q\,  <ft,  q3  it  will  not  be  necessary  to  recalcu- 
late the  values  of  2',  2",  2'",  when  these  have  been  calculated  from  fairly  good  values  of  q\,q2,q3. 
But  when,  as  is  generally  the  case,  the  first  assumption  is  only  a  rude  guess,  the  values  of  2',  2", 
2'"  should  be  recalculated  after  one  or  two  corrections  of  q}  ,  q2,  q3.  To  get  the  best  results  when 
we  do  not  recalculate  2',  2",  2'",  we  may  proceed  as  follows:  Let  2',  2",  2'"  denote  the  values 
which  have  been  calculated;  Dqt,  Dqi,  Dq3,  respectively,  the  sum  of  the  corrections  of  each  of 
the  quautities  q}  ,  qt,  <fr,  which  have  been  made  since  the  calculation  of  2',  2",  2'"  ;  2  the  residual 
after  all  the  corrections  of  qt,  qt,  q3,  which  have  been  made;  and  Jft,  J<fe,  dq3  the  remaining 
corrections  which  we  are  seeking.  We  have,  .then,  very  nearly 


The  same  considerations  which  we  applied  to  equation  (l'l)  enable  us  to  simplify  this  equation 
also,  and  to  write  with  a  fair  degree  of  accuracy 

(30) 

(31) 
where 


•In  the  eaae  mentioned  in  the  preceding  font-in<ti-.  fn«m  [^ga]=— .079  and  pt—.'ift,  we  get  Jg,=— .ll.V., 
in  •ennilily  the  ume  value  M  that  obtained  lir  ralrulntinu  tin-  quadratic  t<-nn. 


MEMOIKS  OF  THE  NATIONAL  ACADEMY  OF  SCIENCES.  87 

CORRECTION   OF   THE   FUNDAMENTAL  EQUATION. 

When  we  have  thus  determined,  by  the  numerical  solution  of  our  fundamental  equation, 
approximate  values  of  the  three  positions  of  the  body,  it  will  always  be  possible  to  apply  a  small 
numerical  correction  to  the  equation,  so  as  to  make  it  agree  exactly  with  the  laws  of  elliptic 
motion  in  a  fictitious  case  differing  but  little  from  the  actual.  After  such  a  correction,  the  equa- 
tion will  evidently  apply  to  the  actual  case  with  a  much  higher  degree  of  approximation. 

There  is  room  for  great  diversity  in  the  application  of  this  principle.  The  method  which 
appears  to  the  writer  the  most  simple  and  direct  is  the  following,  in  which  the  correction  of  the 
intervals  for  aberration  is  combined  with  the  correction  required  by  the  approximate  nature  of 
the  equation.* 

The  solution  of  the  fundamental  equation  gives  us  three  points,  which  must  necessarily  lie  in 
one  plane  with  the  sun,  and  in  the  lines  of  sight  of  the  several  observations.  Through  these  points 
we  may  pass  an  ellipse,  and  calculate  the  intervals  of  time  required  by  the  exact  laws  of  elliptic 
motion  for  the  passage  of  the  body  between  them.  If  these  calculated  intervals  should  be  iden- 
tical with  the  given  intervals,  corrected  for  aberration,  we  would  evidently  have  the  true  solution 
of  the  problem.  But  suppose,  to  fix  our  ideas,  that  the  calculated  intervals  are  a  little  too  long. 
It  is  evident  that  if  we  repeat  our  calculations,  using  in  our  fundamental  equation  intervals  short- 
ened in  the  same  ratio  as  the  calculated  intervals  have  come  out  too  long,  the  intervals  calculated 
from  the  second  solution  of  the  fundamental  equation  must  agree  almost  exactly  with  the  desired 
values.  If  necessary,  this  process  may  be  repeated,  and  thus  any  required  degree  of  accuracy 
may  be  obtained,  whenever  the  solution  of  the  uncorrected  equation  gives  an  approximation  to 
the  true  positions.  For  this  it  is  necessary  that  the  intervals  should  not  be  too  great.  It  appears, 
however,  from  the  results  of  the  example  of  Ceres,  given  hereafter,  in  which  the  heliocentric  mo- 
tion exceeds  62°,  but  the  calculated  values  of  the  intervals  of  time  differ  from  the  given  values  by 
little  more  than  one  part  in  two  thousand,  that  we  have  here  not  approached  the  limit  of  the 
application  of  our  formula. 

In  the  usual  terminology  of  the  subject,  the  fundamental  equation  with  intervals  uncorrected 
for  aberration  represents  the  first  hypothesis,  the  same  equation  with  the  intervals  affected  by  cer- 
tain numerical  coefficients  (differing  little  from  unity)  represents  the  second  hypothesis,  the  third 
hypothesis,  should  such  be  necessary,  is  represented  by  a  similar  equation,  with  corrected  coeffi- 
cients, etc. 

In  the  process  indicated  there  are  certain  economies  of  labor  which  should  not  be  left  un- 
mentioned,  and  certain  precautions  to  be  observed  in  order  that  the  neglected  figures  in  our  com- 
putations may  not  unduly  influence  the  result. 

It  is  evident,  in  the  first  place,  that  for  the  correction  of  our  fundamental  equation  we  need 
not  trouble  ourselves  with  the  position  of  the  orbit  in  the  solar  system.  The  intervals  of  time, 
which  determine  this  correction,  depend  only  on  the  three  heliocentric  distances  r1?  r2,  r3  and  the 
two  heliocentric  angles,  which  will  be  represented  by  Vt—Vi  and  v3— v2,  if  we  write  Vi,  v2,  v3  for 
the  true  anomalies.  These  angles  (v2— Vi  and  v3—vt)  may  be  determined  from  rt,  r2,  r3  and  n},  «2, 713, 
and  therefore  from  r^  r2,  r3  and  the  given  intervals.  For  our  fundamental  equation,  which  may  be 
written 

0,  (33) 


indicates  that  we  may  form  a  triangle  in  which  the  lengths  of  the  sides  shall  be  w,ri,  w2r2,  and  n3r3, 
(let  us  say  for  brevity,  sb  s2,  s},)  and  the  directions  of  the  sides  parallel  with  the  three  heliocentric 
directions  of  the  body.  The  angles  opposite  s}  and  s3  will  be  respectively  v3— vt  and  02— v\.  We 
have  therefore,  by  a  well-known  formula, 

tan 


2 

(34) 


'2    '~V"(si 


—  *3 


*  When  an  approximate  orbit  is  known  in  advance,  we  may  correct  the  fundamental  equation  at  once.     The 
formula)  will  be  given  in  the  Summary,  $  XII. 


SH  MK.M01RS  OF  T1IK   NATIONAL   A(  AKK.MV    OK  SCIKNCES. 

As  soon,  therefore,  as  the  solution  of  our  fundamental  equation  has  given  a  sutlicient  approx- 
imation to  tlie  values  of  r,,  »'..,  c,  (say  five-  or  six-figure  values,  if  our  final  result  is  to  he  as  exact 
as  seven-figure  logarithms  ran  make  it),  we  calculate  n,,  »,,  -».,  with  seven figure  logarithms  by 
equations  (2).  and  the  heliocentric  angles  by  equations  (34). 

The  semi-parameter  corresponding  to  these  values  of  the  heliocentric  distances  and  angles  is 
given  by  the  equation 


The  expression  «i— nj+Ha,  which  occurs  iu  the  value  of  the  semi-parameter,  and  the  expres- 
sion nir,— ;t,r,+«3r3,  or  *,—  «j+«3,  which  occurs  both  in  the  value  of  the  semi-parameter  and  in  the 
formula)  for  determining  the  heliocentric  angles,  represent  small  quantities  of  the  second  order  (if 
we  call  the  heliocentric  angles  small  quantities  of  the  first  order),  and  cannot  be  very  accurately 
determined  from  approximate  numerical  values  of  their  separate  terms.  The  first  of  these  quanti- 
ties may,  however,  be  determined  accurately  by  the  formula 

^*2  f*3 

With  respect  to  the  quantity  «i— *j+«3,  a  little  consideration  will  show  that  if  we  are  careful  to 
use  the  same  value  wherever  the  expression  occurs,  both  iu  the  formula?  for  the  heliocentric  angles 
and  for  the  semi-parameter,  the  inaccuracy  of  the  determination  of  this  value  from  the  cause  men- 
tioned will  be  of  no  consequence  in  the  process  of  correcting  the  fundamental  equation.  For, 
although  the  logarithm  of  *i— *2+*3  as  calculated  by  seven  figure  logarithms  from  rt,  r2,  r3  may  be 
accurate  only  to  four  or  five  figures,  we  may  regard  it  as  absolutely  correct  if  we  make  a  very 
small  change  in  the  value  of  one  of  the  heliocentric  distances  (say  r2).  We  need  not  trouble  our- 
selves farther  about  this  change,  for  it  will  be  of  a  magnitude  which  we  neglect  in  computations 
with  seven-figure  tables.  That  the  heliocentric  angles  thus  determined  may  not  agree  as  closely 
as  they  might  with  the  positions  on  the  lines  of  sight  determined  by  the  first  solution  of  the 
fundamental  equation  is  of  no  especial  consequence  in  the  correction  of  the  fundamental  equation. 
which  only  requires  the  exact  fulfillment  of  two  conditions,  viz.,  that  our  values  of  the  heliocen- 
tric distances  and  angles  shall  have  the  relations  required  by  the  fundamental  equation  to  the 
given  intervals  of  time,  aud  that  they  shall  have  the  relations  required  by  the  exact  laws  of 
elliptic  motion  to  the  calculated  intervals  of  time.  The  third  condition,  tlfat  Sone'of  these  values 
shall  differ  too  widely  from  the  actual  values,  is  of  a  looser  character. 

After  the  determination  of  the  heliocentric  angles  aud  the  semi-parameter,  the  eccentricity  ami 
the  true  anomalies  of  the  three  positions  may  next  be  determined,  aud  from  these  the  intervals  of 
time.  These  processes  require  no  especial  notice.  The  appropriate  formula  will  be  given  in  the 
Summary  of  Formula-. 

DETERMINATION  OF  THE  ORBIT   FROM  THE   THREE   POSITIONS  AND   THE   INTERVALS  OF  TIME. 

The  values  of  the  semi-parameter  and  the  heliocentric  angles  as  given  in  the  preceding  para- 
graphs depend  upon  the  quantity  «,—  *2+*3,  the  numerical  determination  of  which  from  »,,  «2,  and  *3 
is  critical  to  the  second  degree  when  the  heliocentric  angles  are  small.  This  was  of  no  conse- 
quence iu  the  process  which  we  have  called  the  cm-rcctiou  of  the  fundamental  fi/Hation.  But  for 
the  actual  determination  of  the  orbit  from  the  positions  given  by  the  corrected  equation — or  by 
the  uncorrecti-d  equation,  when  we  judge  that  to  be  sufficient — a  more  accurate  determination  of 
this  quantity  will  generally  be  necessary.  This  may  be  obtained  in  different  ways,  of  which  the 
following  is  pei  haps  the  most  simple.  Let  us  set 

54  =  5:i-5,.  (37) 

and  «4  for  the  length  of  the  vector  2«,  obtained  by  taking  the  square  root  of  the  sum  of  the  squares 
of  the  components  of  the  vector.  It  is  evident  that  xt  in  the  longer  aud  «4  the  shorter  diagoual  of 


MEMOIRS  OF  THE  NATIONAL  ACADEMY  OF  SCIENCES.  89 

a  parallelogram  of  which  the  sides  are  «,  and  «3.  The  area  of  the  triangle  having  the  sides  «„  «2,  *3 
is  therefore  equal  to  that  of  the  triangle  having  the  sides  *,,  «3,  «4,  each  being  one-half  of  the 
parallelogram.  This  gives 


«4-«3),   (38) 

and 

S.-S  +8  = 


The  numerical  determination  of  this  value  of  *!—  «2+s3  is  critical  only  to  the  first  degree. 

The  eccentricity  ami  the  true  anomalies  may  be  determined  in  the  same  way  as  in  the  correc- 
tion of  the  formula.  The  position  of  the  orbit  in  space  may  be  derived  from  the  following  consid- 
erations. The  vector  —  S2  is  directed  from  the  sun  toward  the  second  position  of  the  body  ;  the 
vector  (£4  from  the  first  to  the  third  position.  If  we  set 


the  vector  <£5  will  be  in  the  plane  of  the  orbit,  perpendicular  to  —  <£2  and  on  the  side  toward 
which  anomalies  increase.  If  we  write  «5  for  the  length  of  ©5, 

-??  and  ®? 

«2  «5 

will  be  unit  vectors.  Let  3  and  3'  be  unit  vectors  determining  the  position  of  the  orbit,  3  being 
drawn  from  the  sun  toward  the  perihelion,  and  3'  at  right  angles  to  3,  in  the  plane  of  the  orbit, 
and  on  the  side  toward  which  anomalies  increase.  Then 

3=  -cos  v2??-sin  t>2^  (41) 

*2  *5 

3'  =—  sin  r2—  2+cos  e^?  (42) 

*  «2  *5 

The  time  of  perihelion  passage  (T)  may  be  determined  from  any  one  of  the  observations  by 
the  equation 

-.(t—T)=E-e  &in  E,  (43) 

a* 

the  eccentric  anomaly  E  being  calculated  from  the  true  anomaly  v.  The  interval  t—  T  in  this 
equation  is  to  be  measured  in  days.  A  better  value  of  T  may  be  found  by  averaging  the  three 
values  given  by  the  separate  observations,  with  such  weights  as  the  circumstances  may  suggest. 
But  any  considerable  differences  in  the  three  values  of  T  would  indicate  the  necessity  of  a  second 
correction  of  the  formula,  and  furnish  the  basis  for  it. 
For  the  calculation  of  an  ephemeris  we  have 

«R  =  -ae3+cos  E  «3+sin  E  by  (44) 

in  connection  with  the  preceding  equation. 

Sometimes  it  may  be  worth  while  to  make  the  calculations  for  the  correction  of  the  formula 
in  the  slightly  longer  form  indicated  for  the  determination  of  the  orbit.  This  will  be  the  case 
when  we  wish  simultaneously  to  correct  the  formula  for  its  theoretical  imperfection,  and  to  correct 
the  observations  by  comparison  with  others  not  too  remote.  The  rough  approximation  to  the  orbit 
given  by  the  uncorrected  formula  may  be  sufficient  for  this  purpose.  In  fact,  for  observations 
separated  by  very  small  intervals,  the  imperfection  of  the  uncorrected  formula  will  be  likaly  to  af- 
fect the  orbit  less  than  the  errors  of  the  observations. 

The  computer  may  prefer  to  determine  the  orbit  from  the  first  and  third  heliocentric  positions 
with  their  times.  This  process,  which  has  certain  advantages,  is  perhaps  a  little  longer  than 


90  Mi:.M«iil;s  ()1    THE  NATIONAL  ACADEMY  OF  SCIENCES. 

that  here  given,  and  does  not  lend  itself  quite  so  readily  to  successive  improvements  of  the 
hypothesis.  When  it  is  desired  to  derive  an  improved  hypothesis  from  an  orbit  thus  determined, 
the  formula)  in  §  XII  of  the  summary  may  be  used. 

• 

SUMMARY  OF  FORMULA 

WITH  DIRECTIONS  FOB  USB. 
[For  the  case  in  which  an  approximate  orbit  is  known  in  advance,  see  XII.] 

I. 
Preliminary  computations  relating  to  the  intervals  of  time. 

(„  tt,  ?3=times  of  the  observations  in  days. 
log  fc=8.2355814    (after  Gauss) 
T1=k(t3-tt)  r3=fc(<z-*i) 

A  _<3-*»  A  _t*-tt 

'-  '- 


p_-riTi->  R..nT1r3T3  D 

12  ~T2~ 


Forcoutrol:  Ai 

II. 

Preliminary  computations  relating  to  the  first  observation. 

Xi,  FI,  Z\  (components  of  Gi)=the  heliocentric  coordinates  of  the  earth,  increased  by  the  geocen- 

tric coordinates  of  the  observatory. 

fi,  »7»  C:  (components  of  &)=the  direction-cosines  of  the  observed  position,  corrected  for  tin- 

aberration  of  the  fixed  stars. 


Preliminary  computations  relating  to  the  second  and  third  observations. 

The  fin-mill.  e  are  entirely  analogous  to  those  relating  to  the  first  observation,  the  quantities 
being  distinguished  by  the  proper  suffixes. 

III. 

Equations  of  the  first  hypothesis. 

When  the  preceding  quantities  have  been  computed,  their  numerical  values  (or  their  loga- 
rithms, when  more  convenient  for  computation,)  are  to  be  substituted  in  the  following  equations: 


rmnponents  of  5 


For  control: 


MEMOIRS  OF  THE  NATIONAL  ACADEMY  OF  SCIENCES.  91 

Components  of  2' 


Components  of  ( 


For  control : 

-  *r  --*      i    f~  f,      ift,   \  • 

Components  of  ©•• 
P"=-   °™" 


Components  of 


»3 
For  control : 


Components  of  (g'" 
3//,        3R3q3 


The  computer  is  now  to  assume  any  reasonable  values  either  of  the  geocentric  distances,  p,, 
p2,  p3,  or  of  the  heliocentric  distances,  r,,  r2,  r3,  (the  former  in  the  case  of  a  comet,  the  latter  in  the 
case  of  an  asteroid,)  and  from  these  assumed  values  to  compute  the  rest  of  the  following  quantities: 

By  equations  III,,  III'.  By  equations  III.,,  III'.  By  equations  III3,  III'". 

9i  <fe  qa 

'°g  r\  log  r2  log  r3 

l°S  #1  log  JRz  log  R3 

log  (1+^,)  log  (!-#,)  log  (i+R3) 

log  P'  log  P"  log  P"> 

&  A  A 

a!  a"  a1" 

y'  y"  y>» 


92                         MK.MOII.'S  OK  Till-:   NATIONAL   ACADKMY  OF  S(  'I  KNCKS. 

IV. 

Calculations  relating  to  differential  coefficients. 

Components  of  2"  X  2'"                  Components  of  2'"  X  2'  Components  of  2'  X  2" 

at=ft"y'"—y"ft>"                        at=fl'"y'-yl"fl'  a3=/3'y"-y'ft" 

bt=y"a"'-a"y"'                         bt=y'"i*'-a"'y'  b3=y'a"—a'y" 

Cl=a"ftl"-P"a'"                        Ct=a"'/3'-/3'"al  c3 


These  computations  are  controlled  by  the  agreement  of  the  three  values  of  0. 
The  following  are  not  necessary  except  when  the  corrections  to  be  made  are  large  : 


V. 

Corrections  of  the  geocentric  dixtmices. 
Component*  of  2. 


i- 

y=y}+yi+y3  ft« 


(This  equation  will  generally  be  most  easily  solved  by  repeated  substitutions.) 


VI. 

Successive  corrections. 

J^u  Jgz,  J^3  are  to  be  added  as  corrections  to  </,,  172,  «/3.    With  the  new  values  thus  obtained 
the  computation  by  t-cjuations  III,,  111^,  III.,  are  to  be  recoin  .....  need.    Two  courses  are  now  open: 

(a)  The  work  may  be  carried  on  exactly  as  before  to  the  determination  of  new  corrections  for 
0i  >  ft.ft. 

(b)  Tin-  commutations  by  equations  III',  III",  III'",  and  IV  may  be  omitted,  and  the  old  valm  s 
of  a,  ,  ft,,  ci,  flj,  etc.,  O,  and  L  may  be  used  with  the  new  residuals  «v  /?,  y  to  get  new  corrections 
for  91,  9>,  q3  by  the  equations 

0, 


where  Dqt  denotes  the  fornirr  correction  of  </2.     (More  generally,  at  any  stage  of  the  work,  Dqt  will 
n-prcwnt  the  sum  of  all  tin-  cm-icctions  of  </.,  which  have  been  made  since  the  last  computation  of 

<I|  ,*!,<•'• 

So  far  aa  any  general  rule  can  In-  ^-ivcn,  it  is  advised  to  recompute  a,,  6,,  etc.,  and  G  once, 
after  the  second  ••<>!  ivi-timi-  of  ./i  •  '/••  '/  •  unless  tli«'  assumed  values  represent  a  fair 


MEMOIRS  OF  THE  NATIONAL  ACADEMY  OF  SCIENCES.  93 

approximation.    Whether  L  is  also  to  be  recomputed,  depends  on  its  magnitude  and  on  that  of 
the  correction  of  g2,  which  remains  to  be  made.    In  the  later  stages  of  the  work,  when  the  cor- 
rections are  small,  the  terms  containing  L  may  be  neglected  altogether. 
The  corrections  of  qlt  q2j  q3  should  be  repeated  until  the  equations 

«=0  /J=0  y=Q 

are  nearly  satisfied.    Approximate  values  of  rt,  r2,  r3  may  suffice  for  the  following  computations, 
which,  however,  must  be  made  with  the  greatest  exactness. 

VII. 

Test  of  the  first  hypothesis. 
log  rt ,  log  r2,  log  r3,  (approximate  values  from  the  preceding  computations.) 


The  value  of  s— s2  may  be  very  small,  and  its  logarithm  in  consequence  ill  determined.  This 
will  do  no  harm  if  the  computer  is  careful  to  use  the  same  value — computed,  of  course,  as  carefully 
as  possible — wherever  the  expression  occurs  in  the  following  formulae. 


tan  £(v2— t>i  )=—^- 


tan  £(tf3_t>2= 


N 

t 
tan  %(v3— Vi)=- 

For  adjustment  of  values:  £(«3- 

.   e  sin 


U  sin  *(«•)— 1 


e  cos 


•2,  COS  J(t>a— 1 

tan 


For  control :  e  cos  <v2=-—  1 

»*2 


a-  * 


tan  J.Z7i=£  tan  Jv,  tan  JjE72=e  tan  £02  tan  ^E3—f  tan 

sin  _E2— ea$  sin  -&3 
!  sin  E,—ea$  sin  R 


94  MEMOIltS  OF  T11E  NATIONAL  ACADEMY  OF  SCIENCES. 

VIII. 

For  the  second  hypothesis. 

<yn=.0057613il-(/3z-p3)  (aberration-constant  after  Struve.) 

«yr,=.0057613A-(A-Pz)  log  (.0057613fc) =5.99610 

A  log  r,=log  r,-log  (r,  ..i^— rfr,) 
%A  log  r3=log  T3— log  (r3c»ic.— tfr3) 

A  log  (r,r3)=J  log  TI+ A  log  T3 
A  log  — '= J  log  r,— A  log  rs 


J  lOg  £,=J 

J  log  £,= J  log 
J  log  B3= J  log 

These  corrections  are  to  be  added  to  the  logarithms  of  At,  A3,  BI,  B2,  B3,  in  equations  IIIi, 
III,,  IIIj,  and  the  corrected  equations  used  to  correct  the  values  of  qit  g3,  g3,  until  the  residuals  a, 
ft,  y  vanish.  The  new  values  of  At,  A3  must  satisfy  the  relation  Ai+A3=;l.,  and  the  corrections 
J  log  Att  J  log  A3  must  be  adjusted,  if  necessary,  tor  this  end. 

Third  hypothesis. 

A  second  correction  of  equations  IIIi,  III2,  III3  may  be  obtained  in  the  same  manner  as  the 
first,  but  this  will  rarely  be  necessary. 

IX. 

Determination  of  the  ellipse. 
It  is  supposed  that  the  values  of 

«1,    fil,    Pi,  «2,    ftt,     Y*l  «3>    /*3»    K3> 

r\,    r2,    r3,  Ti",,  Kj,  B3,  «,,    «2,    «3, 

have  been  computed  by  equations  III,,  IIIZ,  III3  with  the  greatest  exactness,  so  as  to  make  the 
residuals  a,  ft,  y  vanish,  and  that  the  two  formulae  for  each  of  the  quantities  *,,  «z,  «3  give  sensibly 
the  same  value. 

Components  of  24  Components  of  25 


MEM01ES  OF  THE  NATIONAL  ACADEMY  OF  SCIENCES.  95 

For  control  only  :  ? 


*(*-*,)  (*-«3) 

tan  £(1?,-*,)= 
tan  ir,-tr2= 


K« 


The  computer  should  be  careful  to  use  the  corrected  values  of  AI,  A3.    (See  VUL)    Trifling 
errors  in  the  angles  should  be  distributed. 


PP 

e  sin 


0   . 


.      , 
sin     t?3-i 


e  cos  =     r' 


, 
2  cos  ^(t>3—  t 

tan  ^r3+ri)  e» 


p 

For  control:  ecosrz=—  —  1 


Direction-cosines  of  semi-major-axis. 
,__cosr2     _  sin  r2 

cos  r2 „      sin  r2/? 
m= fh PS 


«5 


Direction-cosines  of  semi-minor-axis. 


«5 


Components  of  the  aemi-axes. 


MI;.MOII;S  01  Tin-:  NATIONAL  ACADEMY  OK  SCIENCES. 

x. 

Time  of  perihelion  passage. 

Corrections  for  aberration. 
tan  AA'i  —  f  tan  .4r,  6t,  =  —  .00576  13pi 

tail  Jf?,=f  tan  J»,  6t.f=  —  .  0057613  pt 

tan  J-E3=f  tan  4r,  <tt3=-.0057613/>, 

log  .0057613=7.76052 


«,+<«,-  T=*-1a?(.E1-e  sia  1?,) 
tt+6tt-T=k-la*(E1-e  sin  J?2) 
3-e  sin  E3) 


The  threefold  determination  of  T  affords  a  control  of  the  exactness  of  the  solution  of  the 
problem.  If  the  discrepancies  in  the  values  of  T  are  such  as  to  require  another  correction  of  the 
formula-  (a  third  hypothesis),  this  may  be  based  on  the  equations 


A  log  n«=  A  log  T3= 

— 


where  T,,,,  Ttll,  T(i)  denote,  respectively,  the  values  obtained  from  the  first,  second,  and  third 
observations,  and  M  the  modulus  of  common  logarithms. 

XI. 

For  an  ephemeris. 

ft 

—  T)=E—  esin  E 


Heliocentric  co-ordinates.    (Components  of 


x=—  ea,+a,  cos  E+bf  sin  E 
y=—ear+ar  cos  E+b,  sin  E 
z=—ea,+a,  cos  E+b,  sin  E 

These  equations  are  completely  controlled  by  the  agreement  of  the  computed  and  observed 
positions  and  the  following  relations  between  the  constants  : 

a,b.+ajb,+a.b,=0  o,*+a,»+a.»=az  b.'+bf+bf^l-*)* 

XII. 

When  an  approximate  orbit  is  known  in  advance,  we  may  use  it  to  improve  our  fundamental 
equation.  The  following  appears  to  be  the  most  simple  method  : 

Find  theexcentric  anomalies  Et,  E},  #,,  and  the  heliocentric  distances  r,,  r*,  r3,  which  belong 
in  the  approximate  orbit  to  the  times  of  observation  corrected  for  aberration. 

Calculate  Btt  &,,  as  in  §  I,  using  these  corrected  times. 

Determine  A,,  A3  by  the  equation 


.          .  . 

sm  (Et—  E,)—  e  sin  E3+e  aiu  Et    sin  (Et—Et)—e  sin  E^+e  sin  E, 
in  connection  with  the  relation  Ai+A3=l. 


MEMOIRS  OP  THE  NATIONAL  ACADEMY  OP  SCIENCES. 
Determine  B2  so  as  to  make 

B, 


97 


4  siii  kEz— 


sin     E3— 


equal  to  either  member  of  the  last  equation. 

It  is  not  necessary  that  the  times  for  which  H,,  E2,  E3,  r,,  r2,  r3,  are  calculated  should  pre- 
cisely agree  with  the  times  of  observation  corrected  for  aberration.  Let  the  former  be  represented 
by  ti',  t2',  t3',  and  the  latter  by  *,",  £2",  t3"  ;  and  let 

A  log  r,=log  (t3"-tt")-\og  (t3'-t2'), 
A  log  r3=log  (^''_^')_log  (tz'-li1). 


We  may  find  BI,  B3,  AI,  A3,  B2,  as  above,  using  f/,  V,  t3',  and  then  use 
correct  their  values,  as  in  §  VIII. 


Jlogrz  to 


NUMERICAL   EXAMPLE. 


To  illustrate  the  numerical  computations  we  have  chosen  the  following  example,  both  on 
account  of  the  large  heliocentric  motion,  and  because  Gauss  and  Oppolzer  have  treated  the  same 
data  by  their  different  methods. 

The  data  are  taken  from  the  Theoria  Motus,  §  159,  viz: 


Times,  1805,  September  
Longitudes  of  Ceres 

5.  51336 

95°  32'    18".  56 

139.42711 

99°  49'     5".  87 

265.  39813 

118°    5'   28".  85 

Latitudes  of  Ceres  .  ...  

—0°  59'   34".  06 

+7°  16'   36".  80 

+7°  38'   49".  39 

Longitudes  of  the  Earth  
Logs  of  the  Sun's  distance.  .. 

342°  54'    56".  00 
0.0031514 

117°  12'   43".  25 
9.  9929861 

241°  58'   50".  71 
0.  0056974 

The  positions  of  Ceres  have  been  freed  from  the  effects  of  parallax  and  aberration. 


I. 


From  the  given  times  we  obtain  the  following  values: 


Control : 


Numbers. 

Logarithms. 

t>—t, 

133.  91375 

2.  1268252 

(j—  * 

125.97102 

2.  1002706 

<3-tt 

259.  88477 

2.  4147809 

At 

.4847187 

9.  6854897 

At 

.  5152812 

9.  7120443 

ri 

.  3358520 

r. 

.  3624066 

Hi 

9.  C692113 

-B, 

.  3183722 

ft 

9.  562391G 

H.  Mis.  597- 


£r,r3r=2.4959081 


MKM01US  OK  TIIK   NATIONAL  ACADKMY   OF  SCIENCES. 

II. 
From  the  given  positions  we  get : 


loK  -Y, 

i,.,)-, 

9.983.V.1:, 
•J.I711748 
0 

± 

.log-Y, 

11796 

9.  94 

0 

+ 

1"«  -V, 

V 

9.  c.7: 
9.951 
0 

— 

l-K  fi 
'"I!  Ji 

8.9845270 

- 

'"K  £« 

9.22H2738 

7 

log«, 
log  »» 

9.  OV.I0294 
9.94K 

7. 



;i.  10-J»)549 

+ 

log  C> 

9.  124d.-'l:t 

+ 

IVrt, 

— 

.9314993 

^_ 

G3'^3 

.  r,:,;K»304 

— 

.8645336 

+ 

PS 

.1006681 

+ 

J»' 

.  7i:«itW4 

+ 

III. 

The  preceding  computations  furnish  the  numerical  values  for  the  equations  IIIi,  III',  111,, 
III",  IIIa,  III'",  which  follow.  Brackets  indicate  that  logarithms  have  been  substituted  for 
nuuilu-rs. 

\\'o  have  now  to  assume  some  values  for  the  heliocentric  distances  rt,  rt,  r3.  A  mean  propor- 
tional between  the  mean  distances  of  Mars  and  Jupiter  from  tin-  Hun  suggests  itself  as  a  reasonable 
assumption.  In  order,  however,  to  test  the  convergence  of  the  computations,  when  the  assump- 
tions are  not  happy,  we  will  make  the  much  less  probable  assumption  (actually  much  farther 
from  the  truth)  that  the  heliocentric  distances  are  an  arithmetieal  mean  between  the  distant ••  > 
<if  Mars  and  Jupiter.  This  gives  .526  for  the  logarithm  of  each  of  the  distances  r,,  r2,  r3.  From 
these  assumed  values  we  compute  the  first  columns  of  numbers  in  the  three  following  tables. 


qi=pi  —.3874081 


a,=  -[8.C700167]('/,-!t.5901555)(l  +  ^, 
/?,=     [9.G833924](</!+  .0900552)( !+/.',)  >  III, 
7/,  =  -[7.9242047](g,+  .3874081)(l+jR,). 
a!  =-.04C775- [8.67002 J «,-!"«,  \ 
/3'=     .482383+[Q.683M\Ri-P'/ii  V  III' 
r':=-.008399-[7.924L>0]RI-P'ri  ) 


4q, 

—  .66731 

—  .04 

—  .00104:54 

+  .0000006 

9i 

+ 

3.22606 

•S75 

9.61317 

2.5142134 

2.514--M40 

log  r, 

+ 

.  :.,'(iOO 

.  I::i960 

-i791 

.  4'.- 

.  42&.':!77 

log  It, 

8.091V-1 

8.364XU 

i'.i74(l 

,  i;i-:, 

log  (!  +  /.',) 

+ 

.  oor,:):! 

.  0091)34 

.0104199 

.0104010 

log  1" 

+ 

8.  01967 

MM 

H.  :v.i:.7  10 

8.3951457 

«i 

+ 

.30136 

BG06 

.  :!:!'.- 

.:w.  is 

Pi 

+ 

L6I 

1.307304 

1.88 

i.  •:-. 

r\ 

.03072 

.OHUfl 

.034 

.0949011 

a1 

— 

.O.'>0505 

ft' 

+ 

.47K» 

10949 

r' 

~~™ 

.00818 

.007960 

rS=qj>+.  1006681 
^,=[0.3183722]r,-J 


a,=  +  [!».22S27;{SJ((/,+  1.7-Si;> 
/S2=-[9.9»008l)0](.7,-   .03(;i:«l9)(l— /t'2)  ^  HI, 

y,=  -[9.ioi.ii;.-ii!i|!v.-  J 


a"-     .101H51-[0.2L>S27]/i>2+P"a2  \ 
/S"=-.!)77»17-f['.).«»'.)(MIS]/,>,+  /"Vy2  (ill 
y"=-A-CMl  +  \\\.W2C.:,\K.i+l>"yt  ) 


MEMOIES  OP  THE  NATIONAL  ACADEMY  OP  SCIENCES. 


99 


Aqu 

-  .77826 

+  .005042 

+  .0013222 

+  .0000021 

?2 

+ 

3.  34235 

2.  56409 

2.  569132 

2.  5704542 

2.  5704563 

log  r2 

+ 

.  52600 

.412233 

.  4130733 

.4132934 

.  4132937 

log  K, 

+ 

8.  74037 

9.  081673 

9.  0791524 

9.  0784920 

log(l-B,) 

+ 

9.  97543 

9.  944142 

9.  9444866 

9.  94457G1! 

log  P» 

+ 

8.71411 

9.  199120 

9.  1954270 

9.  1944:,98 

a. 

+ 

.81059 

.  638489 

.  6397466 

.640(1760 

A 

3.  05379 

2.  172660 

2.  1787230 

2.  1803116 

r* 

— 

.28858 

.  181843 

.  1825486 

.  1827338 

a" 

+ 

.20182 

.2491854 

ft" 

1.  08177 

1.2018221 

r" 



.  13464 

.  1400944 

q3==p3  -.5599304 
r32=g3«+.7130624 
jR3=[9.5G23916]r3-3 


. 

- 


«3=  -[9.3810737 J  (£,+  1.5798163)  (l+R3)  j 
/?3=     [9.6537308]  (q,-  .4630521)  (1+JJ,)  (  III, 
y3=     [8.83612561(23-f   .5599304)  (l  +  R3)  ) 

a"'=-.240477-i9.38107]JR3-P'"o3^ 
p"=  +.450537+ [9.65373]  Z?3-P'"y93  V  III'" 
y"'=+.068569+[S.83G13]R3-P'";r3  ) 


4qa 

—  .80780 

—  .04055 

+  .0025316 

+  .0000031 

93 

+ 

3.24945 

2.44165 

2.40110 

2.4036316 

2.  4036347 

log  r:, 

+ 

0.  52600 

.  412217 

.4057319 

.  4061394 

.  4061399 

log  JR, 

+ 

7.  98439 

8.  325742 

8.  3451948 

8.  3439733 

log  (1  +  B3) 

+ 

.00417 

.  009099 

.  0095108 

.  0094843 

log  1"" 

+ 

7.  91715 

•  8.357016 

8.3817516 

8.  38019SI3 

ay 

1.  17253 

.987590 

.  9785152 

.9790776 

ft 

+ 

1.  26749 

.  910305 

.  f  924956 

.  8936069 

73 

+ 

.  26373 

.  210171 

.  2075292 

.2076940 

a'" 

.  22847 

.  2222335 

ft'" 

+ 

.44441 

.  4390163 

ylll 

.06690 

.  0650888 

IV. 

The  values  of  a',  ft',  etc.,  furnish  the  basis  for  the  computation  of  the  following  quantities : 


i=-.  01254 
£,  =  +  .01726 
Cj=— .15746 


02= -.03517 
b2=-.  00525 
Ct=-.  08526 


a3=— .07232 
b3=-.  00845 
c3= -.04050 


For_(?  we  get  three  values  sensibly  identical.    Adopting  the  mean,  we  set 

#=.01006. 
We  also  get 

#=-.00998,  i=.02322.* 

V. 

Taking  the  values  of  «i,  «2,  etc.,  from  the  columns  under  IIIi,  II  I2,  III3,  we  form  the  residuals 

«=-.06058,  /5=-.16692,  j/=--05557- 

From  these,  with  the  numbers  last  computed,  we  get 

d=  -.65888,  C2=-.76983,  (73=-.79939, 


*  It  would  have  been  better  to  omit  altogether  the  calculation  of  H  and  L,  if  the  small  value  of  the  latter  could 
have  been  foreseen.  In  fact,  it  will  be  found  that  the  terms  containing  L  hardly  improve  the  convergence,  being 
smaller  than  quantities  which  have  been  neglected.  Nevertheless,  the  use  of  these  terms  in  this  example  will  illus- 
trate a  process  which  in  other  cases  may  be  beneficial. 


]00  MM  MOIL'S  OF  T1IK  NATIONAL    ACADKMY   OK  SCIKNCKS. 

which  might  be  used  as  corrroctions  lor  our  values  of  </,,  </..,  </,.     To  get  more  accurate  values  for 
t  hese  corrections  we  set 


or  J<f2=-. 

which  gives 

J$«—  .77826. 

The  quadratic  term  diminishes  tin-  value  of  Jqt  by  .00843.  Subtracting  the  same  quantity 
from  C|  and  Ct  we  get 

Jgi=—  .66731,  Jg3=  -.80780. 

VI. 

Applying  these  corrections  to  the  values  of  g,,  3,,  </3  we  compute  the  second  numerical  columns 
under  equations  III,,  HI2,and  III3.  We  do  not  go  on  to  the  computations  by  equations  Ill',etc., 
but  content  ourselves  with  the  old  values  of  a,,  61,  etc.,  G>,  and  L,  which  with  the  new  residuals, 

«=—  .012595,  6=.  044949,  ;/=.003012, 

give 

Ci=—  .04567,  C,=.004952,  C,=  —  .04064. 


This  gives 

^(?2 

As  the  term  containing  L  lias  increased  the  value  of  dqt  by  .00009,  we  add  this  quantity  to  C\ 
ami  ('  ,  and  get 

Aqi  =  —.04558,  4q3=  -  .04055. 

With  these  corrections  we  compute  the  third  numerical  columns  under  equations  III],  etc. 
This  time  we  recompute  the  quantities  a7,  etc.,  with  which  we  repeat  the  principal  computations  <>t 
IV,  and  get  the  new  values: 

o,  =  -.0167215  o,=-.0335815  a3=  -.0743299 

&,=  +  .  0149145       bt=—  .0054413  b3=  -.0098825 

c,=—  .1576886       c,=  —  .0779570  c*=  -.0474318 

O=.0090929 

The  quantities  //  and  L  we  neglect  as  of  no  consequence  at  this  stage  of  the  approximation. 
With  these  values  the  new  residuals, 

or=  +  .0002919,  /S=-.0000044,  ^=+.0000288, 

give 

^,  =  0,  =+.  0010434,         J«-(%=+.OOI3L>22,         4q3=C3=  +  .0025316. 

These  correct  i(»n>  furnish  the  basis  for  the  fourth  columns  of  numbers  under  equations  III,, 
etc.,  which  give  the  residuals 

«=+.  0000002,  /*=  +  .  0000009,  >/=  +  .0000001, 

and  the  new  corrcrtioiiM 

3=  +  .  0000031. 


MEMOIRS  OF  THE  NATIONAL  ACADEMY  OP  SCIENCES. 


101 


The  corrected  values  of  ql1  g2,  q3  give 

log  r,=0.4282377,  log  r2=0.4132937, 


log  r3=0.4061399. 


We  have  carried  the  approximation  farther  than  is  necessary  for  the  following  correction  of 
the  formula,  in  order  to  see  exactly  where  the  uncorrected  formula  would  lead  us,  and  for  the 
control  afforded  by  the  fourth  residuals. 

VII. 

The  computations  for  the  test  of  the  uncorrected  formula  (the  tirst  hypothesis)  are  as  follows : 


Number  or  arc. 

Logarithm. 

Number  or  arc. 

Logarithm. 

n 

0.  42S2377 

e 

+ 

8.9025438 

r* 

0.  4132937 

£ 

+ 

9.  9652259 

r3 

0.  4061399 

a 

0.  4419546 

AiBirr* 

+ 

.01174865 

8.  0699879 

tan  1  i'i 

— 

—35°  41'  39".  75 

9.  8563809 

B&i~3 

-(- 

.  11980944 

9.  0784911 

tan  ij  !•„ 

— 

—19°  53'  28".  93 

9.  5584981 

AsB-irz"3 

_^_ 

.  01137670 

8.  0560162 

tan  Jc:i 

— 

—  4°  13'  52".  55 

8.  8691380 

N 

-j. 

.  14293479 

9.  1551380 

tan  $Ei 



—33°  33'  0".  17 

9.  8216068 

81 

-f- 

1.  3308476 

0.  1241283 

tan  lE,a 

— 

—  IB°  28'  6".  35 

9.  5237240 

83 

-j- 

2.  2796616 

0.  3578704 

tan  i_E3 

— 

—  3°  54'  24".  21 

8.  8343639 

83 

8 
8—81 

+ 

1.3417404 
2.  4761248 
1.  1452772 

0.  1276685 
0.  3937725 
0.  0589106 

sin  E\ 
sin  £3 
sin  E3 

— 

—67°  6'  0".  34 
—36°  56'  12".  70 
—  7°  48'  48'  .  42 

9.  9643473 
9.7788272 
9.  1333734 

8  —  8a 

-f 

0.  1964632 

9.  2932812 

8—83 

1.  1343844 

0.  0547602 

e<&  sin  EI 

— 

.3387061 

9.5298230 

R 

_|_ 

9.  5065898 

ea%  sin  E? 



.  2209545 

9.  3443029 

P 

+ 

0.  4391732 

ea$  sin  E3 

.  0499861 

8.  6988491 

tan  i(«3  —  vi) 
tan  i(i>3  —  »3) 

+ 

15°  48'  10".  82 
15°  39'  36".  38 

9.  4518296 
9.  4476792 

a*  (Ei-EJ 

+ 

2.  4226307 

0.  3842872 

tan  i(»s  —  »i) 

31°  27'  47".  20 

9.  7866915 

a3  (Ez—Ev) 

+ 

2.3391145 

0.  3690515 

e  sin  i(»3+"i) 

— 

8.  7099387 

^3  calc. 

_J. 

2.  3048791 

0.  3626482 

e  cos  i(»3-j-i>i) 

-f- 

8.  7872701 

Z"l  C.1C. 

+ 

2.  1681461 

0.  3360885 

tan  i(t>3-j-»i) 

— 

—39°  55'  32".  31 

9.  9226686 

VIII. 

The  logarithms  of  the  calculated  values  of  the  intervals  of  time  exceed  those  of  the  given 
values  by  .0002416  for  the  first  interval  (r3)  and  .0002365  for  the  second  (TI).  Therefore,  since  the 
corrections  for  aberration  have  been  incorporated  in  the  data,  we  set  for  the  correction  of  the 
formula  (for  the  second  hypothesis) 

A  log  r,  =— .0002365  A  log  r3=-.0002416 

This  gives 

A  log  A  !=. 0000026  A  log  A3=— .0000025 

A  log  #!=—  .0004872  A  log  52= -.0004782  A  log  B3=—  .0004665 

The  new  values  of  the  logarithms  of  A\ ,  A3  are 

log  ^,=9.6854923  log  43=9.7120418 


104 


MEMOIRS  OP  THE  NATIONAL  ACADEMY  Ol    SC1KNCKS. 


The  equations  for  an  ephemeris  will  then  be : 

T=180G,  June  23.96378,  Paris  mean  time 
[2.8863140](*-T)=-Ein»«xma.-[4.2216530]  sin  E 

Heliocentric  coordinates  relating  to  the  ecliptic. 
*=  +  .1820765-[0.35302Gl]  cos  £-[0.1827783]  sin  E 
y=  -.1244853+  [0.1878904]  cos  £-[0.3603153]  aiu  E 
*=-.0373987+[9.6656285]  cos  £+[9.3320758]  sin  E 

The  agreement  of  the  calculated  geocentric  positions  with  the  data  is  shown  in  the  following 
table: 


Times,  180f>,  September 

6.51336 

139.  42711 

265.  39813 

Second  hypothesis: 

95°32'18".  88 

99°49'  5".  87 

118°  5'28".52 

0".32 

0".00 

-0".  3:) 

—  0°59'34".  01 

7°16'36".  82 

7°38'49".  34 

errors  ......  ....  .. 

0".05 

0".  02 

—  0".  05 

Third  hypothesis: 

95°32'18".  65 

99°49/  5".  82 

118°  5'28".  79 

0".09 

-0".05 

—0".  06 

latitudes  

—  0°59'34".  04 

7°16'36".  78 

7°38'49".  38 

errors  

0".  02 

-0".02 

-0".01 

The  immediate  result  of  each  hypothesis  is  to  give  three  positions  of  the  planet,  from  which, 
with  the  times,  the  orbit  may  be  calculated  in  various  ways,  and  with  different  results,  so  far  as 
the  positions  deviate  from  the  truth  on  account  of  the  approximate  nature  of  the  hypothesis.  In 
some  respects,  therefore,  the  correctness  of  an  hypothesis  is  best  shown  by  the  values  of  the  goo- 
centric  or  heliocentric  distances  which  are  derived  directly  from  it.  The  logarithms  of  the  helio- 
centric distances  are  brought  together  in  the  following  table,  and  corresponding  values  fiom 
Gauss*  and  Oppolzerf  are  added  for  comparison.  It  is  worthy  of  notice  that  the  positions  given 
by  our  second  hypothesis  are  substantially  correct,  and  if  the  orbit  had  been  calculated  1'roni  the 
first  and  third  of  these  positions  with  the  interval  of  time,  it  would  have  left  little  to  be  desired. 


log  r,  . 

logr,. 

log  r,  . 

First  hypothesis  .......... 

.4282377 

4i:fc*i:!7 

40613'  *l 

.42H-,'7--> 

.413 

1061996 

Third  hypothesis  

.42K 

.4132808 

.  4IHJ2003 

Gauss: 
First  hypothesis  ... 

4323934 

.4114720 

4094712 

St-ciuid  hypothesis.... 
Third  hypothesis  
Fourth  hypothesis  

.  I-.-.H773 
.  rj-i-ii 

.  42827'.« 

.4129371 
.4132107 
.41:12317 

.  407197.r> 

.4lHJ4ii'.l7 
.4062033 

OppolziT: 
Firnt  hypothesis 

4281340 

413330 

I  ur,  u  ','.»;» 

Second  h.vpnth- 
Third  hypothesis  

.  4282794 
.  42- 

.4132801 

.40til'.l7li 
.  4/H52009 

fn  comparing  the  different  methods,  it  should  be  observed  that  the  determination  of  the  posi- 
tions in  any  hypothesis  by  (lauss's  method  requires  successive  cot  red  ions  of  a  single  independent 
variable,  a  corresponding  determination  by  ( >ppolzer'a  method  requires  the  successive  corrections 
of  two  independent  variables,  while  the  eonvsp<inding  determination  by  the  method  of  the  pres- 
ent paper  requires  the  successive  corrections  of  three  independent  variables. 

•  Theoria  mot.  ,liiil)rHtiiiinnni«  <|.T  K»iin'tru  niul  Ham-lm,  2d  ed.,  vol.  J,  p.  394. 


